Math Quiz
Find the only prime number in the form of [tex]n^3+1[/tex].
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A number is prime if it is divisible by itself and 1.
We have been given, n³ + 1 to be, there exist only one value.
n³ + 1
On factorising,
= n³ + 1³
Using basic indentity a³ + b³ = (a+b)(a² - ab + b²)
or, n³ + 1³ = (n + 1)(n² − n + 1)
for n to be prime, either (n + 1) = 1 or (n² - n + 1) = 1
n + 1 = 1 is not possible, ∀n∈N
or, (n² - n + 1) = 1
or, n² - n + 1 - 1 = 0
or, n(n-1) = 0
n = 0 or n - 1 = 0 i.e n = 1
neglecting n=0 as 1 is not a prime.
if n = 1,
n³ + 1 = 1 + 1 = 2, which is obviously a prime.
Therefore, in the form, n³+1, the only prime number that exists is 2.
Answer:
It can be prime only when n=30
And,9001 is a prime number.